A Theory Of Everything?
Page 8
DUALITY
The key tool to understanding this breakthrough is something "duality."
Loosely speaking, two theories are "dual" to each other if they can be
shown to be equivalent under a certain interchange.
The simplest example of duality is reversing the role of electricity and
magnetism in the equations discovered by James Clerk Maxwell of
Cambridge University 130 years ago. These are the equations which govern
light, TV, X-rays, radar, dynamos, motors, transformers, even the
Internet and computers. The re- markable feature about these equations
is that they remain the same if we interchange the magnetic B and
electric fields E and also switch the electric charge e with the
magnetic charge g of a magnetic "monopole": E <--> B and e <--> g
(In fact, the product eg is a constant.) This has important
implications. Often, when a theory cannot be solved exactly, we use an
approximation scheme. In first year calculus, for example, we recall
that we can approximate certain functions by Taylor's expansion.
Similarly, since e^2 = 1/137 in certain units and is hence a small
number, we can always approximate the theory by power expanding in e^2.
So we add contributions of order e^2 + e^4 + e^6 etc. in solving for,
say, the collision of two parti- cles. Notice that each contribution is
getting smaller and small- er, so we can in principle add them all up.
This generalization of Taylor's expansion is called "perturbation
theory," where we perturb the system with terms containing e^2.
(For example, in archery, perturbation theory is how we aim our arrows.
With every motion of our arms, our bow gets closer and closer to
aligning with the bull's eye.)
But now try expanding in g^2. This is much tougher; in fact, if we
expand in g^2, which is large, then the sum g^2 + g^4 + g^6 etc. blows
up and becomes meaningless. This is the reason why the
"non-perturbative" region is so difficult to probe, since the theory
simply blows up if we try to naively use perturbation theory for large
coupling constant g. So at first it appears hopeless that we could ever
penetrate into the non-perturbative region.
(For example, if every motion of our arms got bigger and bigger, we
would never be able to zero in and hit the target with the arrow.)
But notice that because of duality, a theory of small e (which is easily
solved) is identical to a theory of large g (which is difficult to
solve). But since they are the same theo- ry, we can use duality to
solve for the non-perturbative region.
S,T, AND U DUALITY
The first inkling that duality might apply in string theory was
discovered by K. Kikkawa and M. Yamasaki of Osaka Univ. in 1984. They
showed that if you "curled up" one of the extra dimen- sions into a
circle with radius R, the theory was the same if we curled up this
dimension with radius 1/R. This is now called T- duality: R <--> 1/R
When applied to various superstrings, one could reduce 5 of the string
theories down to 3 (see figure). In 9 dimensions (with one dimension
curled up) the Type IIa and IIb strings were iden- tical, as were the
E(8)xE(8) and O(32) strings.
Unfortunately, T duality was still a perturbative duality. The next
breakthrough came when it was shown that there was a second class of
dualities, called S duality, which provided a duality between the
perturbative and non-perturbative regions of string theory. Another
duality, called U duality, was even more powerful.
Then Nathan Seiberg and Witten brilliantly showed how anoth- er form of
duality could solve for the non-perturbative region in four dimensional
supersymmetric theories.
However, what finally convinced many physicists of the power of this
technique was the work of Paul Townsend and Edward Wit- ten. They caught
everyone by surprise by showing that there was a duality between 10
dimensional Type IIa strings and 11 dimension- al supergravity! The
non-perturbative region of Type IIa strings, which was previously a
forbidden region, was revealed to be governed by 11 dimensional
supergravity theory, with one dimen- sion curled up.
At this point, I remember that many physicists (myself included) were
rubbing our eyes, not believing what we were seeing. I remember saying
to myself, "But's that's impossible!"
All of a sudden, we realized that perhaps the real "home" of string
theory was not 10 dimensions, but possibly 11, and that the theory
wasn't fundamentally a string theory at all! This revived tremendous
interest in 11 dimensional theories and p- branes. Lurking in the 11th
dimension was an entirely new theory which could reduce down to 11
dimensional supergravity as well as 10 dimensional string theory and
p-brane theory.
DETRACTORS OF STRING THEORIES
To the critics, however, these mathematical developments still don't
answer the nagging question: how do you test it? Since string theory is
really a theory of Creation, when all its beautiful symmetries were in
their full glory, the only way to test it, the critics wail, is to
re-create the Big Bang itself, which is impossible. Nobel Laureate
Sheldon Glashow likes to ridicule superstring theory by comparing it
with former Pres. Reagan's Stnagging question: how do you test it? Since
string theory is really a theory of Creation, when all its beautiful
Actually, most string theorists think these criticisms are silly. They
believe that the critics have missed the point.
The key point is this: if the theory can be solved non- perturbatively
using pure mathematics, then it should reduce down at low energies to a
theory of ordinary protons, electrons, atoms, and molecules, for which
there is ample experimental data. If we could completely solve the
theory, we should be able to extract its low energy spectrum, which
should match the familiar particles we see today in the Standard Model.
Thus, the problem is not building atom smashers l,000 light years in
diameter; the real problem is raw brain power: of only we were clever
enough, we could write down M-theory, solve it, and settle everything.
EVOLVING BACKWARDS
So what would it take to actually solve the theory once and for all and
end all the speculation and back-biting? There are several approaches.
The first is the most direct: try to derive the Standard Model of
particle interactions, with its bizarre collection of quarks, gluons,
electrons, neutrinos, Higgs bosons, etc. etc. etc. (I must admit that
although the Standard Model is the most successful physical theory ever
proposed, it is also one of the ugliest.) This might be done by curling
up 6 of the 10 dimensions, leaving us with a 4 dimensional theory that
might resemble the Standard Model a bit. Then try to use duality and M-
theory to probe its non-perturbative region, seeing if the symme- tries
break in the correct fashion, giving us the correct masses of the quarks
and other particles in the Standard Model.
Witten's philosophy, however, is a bit different. He feels that the key
to solving string theory is to understand the under- lying principle
behind the theory.
Let me explain. Einstein's theory of general relativity, for example,
started from first principles. Einstein had the "happi- est thought in
his life" when he leaned back in his chair at the Bern patent office and
realized that a person in a falling eleva- tor would feel no gravity.
Although physicists since Galileo knew this, Einstein was able to
extract from this the Equivalence Principle. This deceptively simple
statement (that the laws of physics are indistinguishable locally in an
accelerating or a gravitating frame) led Einstein to introduce a new
symmetry to physics, general co-ordinate transformations. This in turn
gave birth to the action principle behind general relativity, the most
beautiful and compelling theory of gravity. Only now are we trying to
quantize the theory to make it compatible with the other forces. So the
evolution of this theory can be summarized as: Principle -> Symmetry ->
Action -> Quantum Theory
According to Witten, we need to discover the analog of the Equivalence
Principle for string theory. The fundamental problem has been that
string theory has been evolving "backwards." As Witten says, "string
theory is 21st century physics which fell into the 20th century by
accident." We were never "meant" to see this theory until the next
century.
IS THE END IN SIGHT?
Vafa recently added a strange twist to this when he intro- duced yet
another mega-theory, this time a 12 dimensional theory called F-theory
(F for "father") which explains the self-duality of the IIb string.
(Unfortunately, this 12 dimensional theory is rather strange: it has two
time co-ordinates, not one, and actu- ally violates 12 dimensional
relativity. Imagine trying to live in a world with two times! It would
put an episode of Twilight Zone to shame.)
So is the final theory 10, 11, or 12 dimensional?
Schwarz, for one, feels that the final version of M-theory may not even
have any fixed dimension. He feels that the true theory may be
independent of any dimensionality of space-time, and that 11 dimensions
only emerges once one tries to solve it. Townsend seems to agree, saying
"the whole notion of dimensional- ity is an approximate one that only
emerges in some semi-classi- cal context."
So does this means that the end is in sight, that we will someday soon
derive the Standard Model from first principles?
I asked some of the leaders in this field to respond to this question.
Although they are all enthusiastic supporters of this revolution, they
are still cautious about predicting the future.
Townsend believes that we are in a stage similar to the old quantum era
of the Bohr atom, just before the full elucidation of quantum mechanics.
He says, "We have some fruitful pictures and some rules analogous to the
Bohr-Sommerfeld quantization rules, but it's also clear that we don't
have a complete theory."
Duff says, "Is M-theory merely a theory of supermembranes and super
5-branes requiring some (as yet unknown) non- perturbative quantization,
or (as Witten believes) are the under- lying degrees of freedom of
M-theory yet to be discovered? I am personally agnostic on this point."
Witten certainly believes we are on the right track, but we need a few
more "revolutions" like this to fi"revolutions" like this to finally
solve the theory. "I think there are still a couple more superstring
revo- lutions in our future, at least.
Vafa says, "I hope this is the 'light at the end of the tunnel' but who
knows how long the tunnel is!"
Schwarz, moreover, has written about M-theory: "Whether it is based on
something geometrical (like supermembranes) or some- thing completely
different is still not known. In any case, finding it would be a
landmark in human intellectual history."
Personally, I am optimistic. For the first time, we can see the outline
of the lion, and it is magnificent. One day, we will hear it roar.
BEFORE THE BIG BANG
What's the farthest object in the universe?
I am sometimes asked that age-old question to kick off discussion during
a grueling 15-city tour lecturing about my book, Hyperspace.
I point out that with the naked eye, one can easily see out to several
hundred light years, the distance to the flickering stars making up the
dazzling firmament on a clear night. In fact, the seemingly "infinite"
heavenly display we see makes up only the tiniest wrinkle in the Orion
arm of the Milky Way galaxy.
With a pair of binoculars turned on the Milky Way itself, a dim, white
haze becomes a brilliant sheet of stars which are tens of thousands of
light years away.
With the world's most powerful telescopes, you can detect the quasars.
Because of their enormous redshift, we estimate that they lie billions
of light years away, close to the very edge of the visible universe.
At even farther distances, we are peering into Creation itself. In 1992
the COBE satellite allowed astronomers to carry out detailed
measurements on the "echo of Creation," the 3 degree microwave radiation
that uniformly fills up the universe. This ancient, relic radiation,
older than the stars themselves, dates back to just 300,000 years after
the Big Bang, which took place perhaps 15 to 20 billion years ago.
But without fail, someone in the audience then asks the
innocent-sounding question, "But professor, what happened before the Big
Bang?"
At this point, I usually detect a faint, satisfied smirk developing on
the faces of a few people in the audience, as if they have finally
stumped the lecturer. I know that they expect me to throw up my hands,
gaze glassy-eyed into the heavens, and sigh philosophically, "We
scientists just don't know. We don't even have a clue. It's one of the
great unanswered mysteries of nature. Perhaps we'll never know."
Actually, I see a lot of startled faces when I reply, "I'm glad you
asked, because that is the subject of today's lecture. Today, we will
discuss what probably happened before Creation. Analyzing this question
is what I do for a living."
QUANTUM COSMOLOGY: A NEW SCIENCE IS BORN
What catches them off guard is that in the leading physics laboratories
around the world, the universe before the Big Bang has become one of the
hottest areas of research. There is a tangible air of excitement and
anticipation as we witness the birth of a new science called "quantum
cosmology." Although there is no experimental proof for quantum
cosmology, the theory is so compelling and beautiful that it has become
the center of intense research. Already, the theory has forced us,
almost against our will, to confront the bizarre possibility of parallel
universes, wormholes, and the 10th dimension. Many physicists are
leaping into this game, following the lead of such pioneers as Stephen
Hawking and Nobel laureate Murray Gell-Mann.
At first, "quantum cosmology" appears to be an oxymoron, a contradiction
in terms. After all, cosmology is based on Einstein's general theory of
relativity, a theory of gravity which compares the expanding universe to
a smooth balloon being inflated by a child, with trillions of tiny
galaxies sprinkled on the surface like star dust. By contrast, the
quantum theory refers to the sub-atomic world, populated by thousands of
strange denizens such as electrons, protons, quarks, and possibly
superstrings.
Like oil and water, general relativity and the quantum theory don't mix.
For example, they take precisely opposite strategies in describing
gravity. General relativity views gravity emerging from the warping of
the continuous, smooth fabric of space-time, while the quantum theory,
by contrast, sees gravity emerging by the exchange of tiny packets of
energy, called "gravitons."
For the past 50 years, there has been a "cold war" between general
relativity and the quantum theory; each theory has developed
independently of the other, and has had unparalleled success as long as
they stayed within their own domain. However, the two theories must
necessarily collide at the instant of the Big Bang, when gravitational
forces and temperatures were so fierce that even particles would have
been ripped apart. At these energies, Einstein's theory becomes useless
and the quantum theory takes over. One can calculate the energy at which
quantum effects overwhelm general relativity, and it is 10^19 billion
electron volts - a quadrillion times greater than the energy of the
canceled supercollider, or SSC. (By comparison, this temperature is a
trillion trillion times greater than that found at the center of a
hydrogen bomb).
In other words, the secret of the origin of the Big Bang lies with
merging the two theories into a higher one, a "theory of everything"
which can explain both theories. What is needed is a quantum theory of
gravity which can simultaneously describe both the sub-atomic quantum
world and the structure of the universe. And this shotgun marriage of
general relativity and the quantum theory is producing even more bizarre
progeny, such as parallel universes and hyperspace.
Sheila Na Gig
Send e-mail to mapona@yahoo.com
Copyright © 1999 Sheila Na Gig.
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